Olympiad problems

2007 Polish MO Finals, 1

Tags: circumcircle , geometry

1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$

1997 IMO Shortlist, 5

Tags: geometry , 3d geometry , tetrahedron , triangle

Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.

2010 LMT, 9

Tags:

Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$

2004 Germany Team Selection Test, 2

Tags: excenter , circumcircle , circumcenter , triangle , geometry

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

1988 Flanders Math Olympiad, 4

Tags: calculus , derivative , geometry , trigonometry , function , inequalities , quadratic

Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad). Prove $2R\theta$ is between $1$ and $\pi$.

2017 China Northern MO, 8

Tags: inequalities

Let $n>1$ be an integer, and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1, x_2, ..., x_n \in [0,n]$ with $x_1x_2...x_n = (n-x_1)(n-x_2)...(n-x_n)$. Find the maximum value of $y = x_1 + x_2 + ... + x_n$.

STEMS 2021-22 Math Cat A-B, A5 B5

Tags: geometry

Let $\triangle ABC$ be an acute angled triangle. Let $G$ be the centroid and let $D$ be the foot of the altitude from $A$ onto $BC$. Let ray $GD$ intersect $(ABC)$ at $X$ and let $AG$ intersect nine point circle at $Y$ not on $BC$. Let $Z$ be the intersection of the $\text{A-tangent}$ to $(ABC)$ and $\text{A-midline}$. Prove that perpendicular from $Z$ to the Euler line, $AX$ and $DY$ concur. The line joining the midpoints of $AB$ and $AC$ is called the $\text{A-midline}$. $(ABC)$ denotes the circumcircle of $\triangle ABC$

2000 Hong kong National Olympiad, 1

Tags: circumcircle , geometry

Let $O$ be the circumcentre of a triangle $ABC$ with $AB > AC > BC$. Let $D$ be a point on the minor arc $BC$ of the circumcircle and let $E$ and $F$ be points on $AD$ such that $AB \perp OE$ and $AC \perp OF$ . The lines $BE$ and $CF$ meet at $P$. Prove that if $PB=PC+PO$, then $\angle BAC = 30^{\circ}$.

2017 Regional Olympiad of Mexico Southeast, 5

Tags: geometry , circumcircle

Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.

2008 District Olympiad, 4

Tags: algebra , polynomial , superior algebra , ring theory , finite field , wedderburn

Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient. [b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $ [b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $

2001 JBMO ShortLists, 3

Tags: modular arithmetic , number theory

Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$.

2010 Polish MO Finals, 2

Tags: function , algebra , polynomial , number theory

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

2023 Paraguay Mathematical Olympiad, 2

Tags: algebra , geometry

Aidée draws ten squares of different sizes. The diagonal of the first square measures $1$ cm, the diagonal of the second measures $2$ cm, the diagonal of the third measures $3$ cm, and so on until the diagonal of the tenth square measures $10$ cm. How much are the areas of the ten squares?

2016 Kosovo National Mathematical Olympiad, 1

Tags: number theory

Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .

2001 China Team Selection Test, 1

Tags: geometry

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

2023 Vietnam National Olympiad, 6

Tags: set theory , combinatorics

There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously. a) Find $m$ when $n = 8, a = 4 , b = 1$ . b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$. c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.

2008 China Team Selection Test, 2

Tags: induction , number theory , algebra , sequence , recurrence relation

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

2022 Regional Olympiad of Mexico West, 5

Tags: floor function , number theory , divide

Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.

2008 Germany Team Selection Test, 3

Tags: polynomial , inequalities , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

2015 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , inequation

Solve the inequation: $$5\mid x\mid \leq x(3x+2-2\sqrt{8-2x-x^2})$$

1986 Vietnam National Olympiad, 1

Tags: function , inequalities

Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]

2016 PUMaC Algebra Individual A, A3

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For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2y}$. The sum of the solutions to the equation \[4096f(f(x, x), x) = x^{13}\] can be written in simplest form as $\tfrac{m}{n}$. Compute $m + n$.

2008 National Olympiad First Round, 16

Tags: inequalities

A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$, at least $2$ of any $40$ gave exactly $2$, and at least $3$ of any $40$ gave exactly $1$ correct answers. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 26 \qquad\textbf{(D)}\ 28 \qquad\textbf{(E)}\ \text{None of the above} $

2020 Jozsef Wildt International Math Competition, W52

Tags: calculus , derivative , inequalities

If $f\in C^{(3)}([0,1])$ such that $f(0)=f(1)=f'(0)=0$ and $|f'''(x)|\le1,(\forall)x\in[0,1]$, show that: a) $$|f(x)|\le\frac{x(1-x)}{\sqrt3}\cdot\left(\int^x_0\frac{f(t)}{t(1-t)}dt\right)^{1/2},(\forall)x\in[0,1]$$ b) $$|f'(x)|\le\frac{1-2x}{\sqrt3}\cdot\left(\int^x_0\frac{|f(t)|}{t(1-t)}dt\right)^{1/2},(\forall)x\in\left[0,\frac12\right]$$ c) $$\int^1_0(1-x)^2\cdot\frac{|f(x)|}xdx\ge9\int^1_0\left(\frac{f(x)}x\right)^2dx$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

MBMT Team Rounds, 2020.7

Tags:

Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? [i]Proposed by Gabriel Wu[/i]